When we talk about probability, it's important to understand two different ideas: theoretical probability and experimental probability. Both help us figure out how likely something is to happen, but they do it in different ways.

Theoretical Probability is all about what we expect to happen based on calculations. It’s like imagining a perfect world. For example, think about flipping a fair coin. The theoretical probability of getting heads is $P(\text{Heads}) = \frac{1}{2}$, or 50%. This is because there are two possible outcomes: heads or tails.

Another example is rolling a six-sided die. The theoretical probability of rolling a 4 is $P(4) = \frac{1}{6}$. That’s because there’s one way to get a 4 out of six possible numbers.

Now, let’s look at Experimental Probability. This is based on real-life tests or experiments. It’s all about what actually happens when you try something. Let’s go back to our coin toss. If you flip the coin 100 times and get heads 47 times, the experimental probability of landing on heads is $P(\text{Heads}) = \frac{47}{100}$, or 47%. This number can be different from the theoretical probability because it shows what really happened.

Another example is rolling a die. If you roll it 60 times and get a '3' only 8 times, then the experimental probability of rolling a 3 is $P(3) = \frac{8}{60} \approx 0.13$, or 13.3%. But from theory, we know that it should be around 16.7% or $\frac{1}{6}$.

Here’s a quick recap:

• Theoretical Probability:

  • Coin toss (heads): $P(\text{Heads}) = \frac{1}{2}$
  • Die roll (4): $P(4) = \frac{1}{6}$

• Experimental Probability (from actual tests):

  • Coin toss (got heads 47 times out of 100 flips): $P(\text{Heads}) = \frac{47}{100}$
  • Die roll (3 showed up 8 times out of 60 rolls): $P(3) = \frac{8}{60}$

So, theoretical probability tells us what should happen in a perfect world, while experimental probability shows us the surprising things that can happen in real life!